Thanks for your wise advice. Could you give me a clear and well-explained way of improving the downsides of my playing? Like, for example, how to improve my phrasing and how to do a smooth rubato. I would like to have a answer for all of them.
There are a few components to this as I see it if you are looking at this long-term. Unfortunately it's difficult to give you a logical step by step that you can take, implement and see direct results. It's easier to point out specific things if you post a longer recording.
1. Develop your ear to hear the qualities you are trying to build into your playing. You can't do what you can't hear. Specifically:
- listen to good singers, good string players, or just any good player on a melody instrument that can do sustained notes (and dynamics). Listen to their phrasing. Try to emulate it, to create the illusion that you are binding notes together the same way, in different breaths or bows (even though the piano in fact is a percussive instrument with immediately decaying sounds). Listen to the different intervals in melodies, their individual qualities, when handled by good musicians. Ultimately the goal of the art is to emulate the human voice with a percussive instrument.
- play ensemble music with singers, string players, and/or other melody instruments for the same reason if you have the opportunity
- keep listening to recordings by famous pianists, try to articulate what's different in recordings which feel more musical and natural to you, versus others that don't
- specifically in your recording for reference, the tone quality between certain melody notes is a bit lumpy/uneven, which hurts the illusion that they are bound together in a phrase.
2. Develop your technique. The better your technique is, the easier it will be for your body to execute what you are hearing in your head and do the musical things we are talking about. Sometimes when you try to get increased control over musical details or force speed in pieces that are too hard, you can start tensing up, but this will only hinder you. When your technique is undeveloped, it will typically be more uncoordinated and tense unless you are one of the lucky few with natural technique. You really benefit from getting a teacher who can teach you relaxed, natural technique which makes it feel easy to play even difficult things.
3. Study theory, harmony. A lot of the "grammar" of western classical music is based on the purpose different chords and intervals had to the composers. Which chords and intervals create tension, and which create release. Which chords are a surprise, and which are expected. The underlying harmony influences the phrase. If you aren't aware of this stuff, you'll miss many things that a proffessional sees and hears, if that's the level you are aiming for.
The quickest short term solution is to bring your piece to a teacher or here and get direct feedback on some things you could improve.
EDIT: another short term solution that might bring improvement is to sing the melodies you are going to play and then try to emulate your own singing as closesly as you can on the piano.